theorem :: LIMFUNC1:99

for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & lim_in-infty f1 = 0 & ( for r being Real ex g being Real st
( g < r & g in dom (f1 (#) f2) ) ) & ex r being Real st f2 | (left_open_halfline r) is bounded holds
( f1 (#) f2 is convergent_in-infty & lim_in-infty (f1 (#) f2) = 0 )